American Institute of Mathematical Sciences

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June  2018, 23(4): 1721-1737. doi: 10.3934/dcdsb.2018073

Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey

Na Min and  Mingxin Wang ,

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Mingxin Wang

Received  August 2017 Published  January 2018

Fund Project: This work was supported by NSFC Grant 11371113.

In this paper, a diffusive prey-predator model with strong Allee effect growth rate and a protection zone $\Omega _0$ for the prey is investigated. We analyze the global existence, long time behaviors of positive solutions and the local stabilities of semi-trivial solutions. Moreover, the conditions of the occurrence and avoidance of overexploitation phenomenon are obtained. Furthermore, we demonstrate that the existence and stability of non-constant steady state solutions branching from constant semi-trivial solutions by using bifurcation theory. Our results show that the protection zone is effective when Allee threshold is small and the protection zone is large.

Keywords: Ratio-dependent prey-predator model, reaction-diffusion, Allee effect, protection zone, bifurcation.
Mathematics Subject Classification: Primary: 92D25, 92D50; Secondary: 35B32, 35B35, 35B40, 35K57.
Citation: Na Min, Mingxin Wang. Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1721-1737. doi: 10.3934/dcdsb.2018073
References:
[1]

W. C. Allee, Principles of Animal Ecology, Saunders, RI, 1949. Google Scholar

[2]

H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent prediction: An abstraction that works, Ecology, 76 (1995), 995-1004.   Google Scholar

[3]

R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551.  doi: 10.2307/1940007.  Google Scholar

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R. G. Casten and C. G. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Diff. Equat., 27 (1978), 266-273.  doi: 10.1016/0022-0396(78)90033-5.  Google Scholar

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R. H. Cui, J. P. Shi and B. Y. Wu, Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Diff. Equat., 256 (2014), 108-129.  doi: 10.1016/j.jde.2013.08.015.  Google Scholar

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Y. H. Du and X. Liang, A diffusive competition model with a protection zone, J. Diff. Equat., 244 (2008), 61-86.  doi: 10.1016/j.jde.2007.10.005.  Google Scholar

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Y. H. Du, R. Peng and M. X. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Diff. Equat., 246 (2009), 3932-3956.  doi: 10.1016/j.jde.2008.11.007.  Google Scholar

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Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Diff. Equat., 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013.  Google Scholar

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S. B. Hsu, On global stability of a predator-prey system, Math. Biosci., 39 (1978), 1-10.  doi: 10.1016/0025-5564(78)90025-1.  Google Scholar

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J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

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C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Ent., 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.  Google Scholar

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T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-New York, 1966.  Google Scholar

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P. E. Kloeden and C. Pötzsche, Dynamics of modified predator-prey models, Int. J. Bifurc. Chaos, 20 (2010), 2657-2669.  doi: 10.1142/S0218127410027271.  Google Scholar

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Y. Lou, Some reaction diffusion models in spatial ecology, Sci. Sin. Math., 45 (2015), 1619-1634.  doi: 10.1360/N012015-00233.  Google Scholar

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Y. Lou and B. Wang, Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment, J. Fixed Point Theory Appl., 19 (2017), 755-772.  doi: 10.1007/s11784-016-0372-2.  Google Scholar

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Y. Li and M. X. Wang, Stationary pattern of a diffusive prey-predator model with trophic intersections of three levels, Nonlinear Anal. RWA, 14 (2013), 1806-1816.  doi: 10.1016/j.nonrwa.2012.11.012.  Google Scholar

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K. Mischaikow, H. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.  doi: 10.1090/S0002-9947-1995-1290727-7.  Google Scholar

[19]

N. Min and M. X. Wang, Qualitative analysis for a diffusive predator-prey model with a transmissible disease in the prey population, Comput. Math. Appl., 72 (2016), 1670-1689.  doi: 10.1016/j.camwa.2016.07.028.  Google Scholar

[20]

L. Nirenberg, Topics in Nonlinear Functional Analysis, Amer. Math. Soc., Providence, RI, 2001. doi: 10.1090/cln/006.  Google Scholar

[21]

W.J. Ni and M. X. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Diff. Equat., 261 (2016), 4244-4272.  doi: 10.1016/j.jde.2016.06.022.  Google Scholar

[22]

W. J. Ni and M. X. Wang, Dynamicl properties of a Leslie-Gower prey-predator model with strong Allee effect in prey, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3409-3420.  doi: 10.3934/dcdsb.2017172.  Google Scholar

[23]

P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 919-942.  doi: 10.1017/S0308210500002742.  Google Scholar

[24]

F. Rao and Y. Kang, The complex dynamics of a diffusive prey-predator model with an Allee effect in prey, Ecol. Complex., 28 (2016), 123-144.  doi: 10.1016/j.ecocom.2016.07.001.  Google Scholar

[25]

M. X. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Phys. D, 196 (2004), 172-192.  doi: 10.1016/j.physd.2004.05.007.  Google Scholar

[26]

M. X. Wang, Nonlinear Elliptic Ppartial Differential Equations (in Chinese), Science Press, Beijing, 2010. Google Scholar

[27]

Y. X. Wang and W. T. Li, Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone, Nonlinear Anal. RWA, 14 (2013), 224-245.  doi: 10.1016/j.nonrwa.2012.06.001.  Google Scholar

[28]

J. F. Wang, J. P. Shi and J. J. Wei, Dynamics and pattern formation in a diffusive predator-prey systems with strong Allee effect in prey, J. Diff. Equat., 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.  Google Scholar

[29]

Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu, The Introduction of Reaction-Diffusion Equations (in Chinese), Science Press, Beijing, 2011. Google Scholar

[30]

F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Equat., 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.  Google Scholar

[31]

J. Zhou and C. L. Mu, Coexistence of a diffusive predator-prey model with Holling type-Ⅱ functional response and density dependent mortality, J. Math. Anal. Appl., 385 (2012), 913-927.  doi: 10.1016/j.jmaa.2011.07.027.  Google Scholar

show all references

References:
[1]

W. C. Allee, Principles of Animal Ecology, Saunders, RI, 1949. Google Scholar

[2]

H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent prediction: An abstraction that works, Ecology, 76 (1995), 995-1004.   Google Scholar

[3]

R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551.  doi: 10.2307/1940007.  Google Scholar

[4]

R. G. Casten and C. G. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Diff. Equat., 27 (1978), 266-273.  doi: 10.1016/0022-0396(78)90033-5.  Google Scholar

[5]

R. H. Cui, J. P. Shi and B. Y. Wu, Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Diff. Equat., 256 (2014), 108-129.  doi: 10.1016/j.jde.2013.08.015.  Google Scholar

[6]

Y. H. Du and X. Liang, A diffusive competition model with a protection zone, J. Diff. Equat., 244 (2008), 61-86.  doi: 10.1016/j.jde.2007.10.005.  Google Scholar

[7]

Y. H. Du, R. Peng and M. X. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Diff. Equat., 246 (2009), 3932-3956.  doi: 10.1016/j.jde.2008.11.007.  Google Scholar

[8]

Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Diff. Equat., 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013.  Google Scholar

[9]

S. B. Hsu, On global stability of a predator-prey system, Math. Biosci., 39 (1978), 1-10.  doi: 10.1016/0025-5564(78)90025-1.  Google Scholar

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[11]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Ent., 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.  Google Scholar

[12]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-New York, 1966.  Google Scholar

[13]

P. E. Kloeden and C. Pötzsche, Dynamics of modified predator-prey models, Int. J. Bifurc. Chaos, 20 (2010), 2657-2669.  doi: 10.1142/S0218127410027271.  Google Scholar

[14]

Y. Lou, Some reaction diffusion models in spatial ecology, Sci. Sin. Math., 45 (2015), 1619-1634.  doi: 10.1360/N012015-00233.  Google Scholar

[15]

Y. Lou and B. Wang, Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment, J. Fixed Point Theory Appl., 19 (2017), 755-772.  doi: 10.1007/s11784-016-0372-2.  Google Scholar

[16]

Y. Li and M. X. Wang, Stationary pattern of a diffusive prey-predator model with trophic intersections of three levels, Nonlinear Anal. RWA, 14 (2013), 1806-1816.  doi: 10.1016/j.nonrwa.2012.11.012.  Google Scholar

[17]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton Univ. Press, Princeton, 1973. Google Scholar

[18]

K. Mischaikow, H. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.  doi: 10.1090/S0002-9947-1995-1290727-7.  Google Scholar

[19]

N. Min and M. X. Wang, Qualitative analysis for a diffusive predator-prey model with a transmissible disease in the prey population, Comput. Math. Appl., 72 (2016), 1670-1689.  doi: 10.1016/j.camwa.2016.07.028.  Google Scholar

[20]

L. Nirenberg, Topics in Nonlinear Functional Analysis, Amer. Math. Soc., Providence, RI, 2001. doi: 10.1090/cln/006.  Google Scholar

[21]

W.J. Ni and M. X. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Diff. Equat., 261 (2016), 4244-4272.  doi: 10.1016/j.jde.2016.06.022.  Google Scholar

[22]

W. J. Ni and M. X. Wang, Dynamicl properties of a Leslie-Gower prey-predator model with strong Allee effect in prey, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3409-3420.  doi: 10.3934/dcdsb.2017172.  Google Scholar

[23]

P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 919-942.  doi: 10.1017/S0308210500002742.  Google Scholar

[24]

F. Rao and Y. Kang, The complex dynamics of a diffusive prey-predator model with an Allee effect in prey, Ecol. Complex., 28 (2016), 123-144.  doi: 10.1016/j.ecocom.2016.07.001.  Google Scholar

[25]

M. X. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Phys. D, 196 (2004), 172-192.  doi: 10.1016/j.physd.2004.05.007.  Google Scholar

[26]

M. X. Wang, Nonlinear Elliptic Ppartial Differential Equations (in Chinese), Science Press, Beijing, 2010. Google Scholar

[27]

Y. X. Wang and W. T. Li, Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone, Nonlinear Anal. RWA, 14 (2013), 224-245.  doi: 10.1016/j.nonrwa.2012.06.001.  Google Scholar

[28]

J. F. Wang, J. P. Shi and J. J. Wei, Dynamics and pattern formation in a diffusive predator-prey systems with strong Allee effect in prey, J. Diff. Equat., 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.  Google Scholar

[29]

Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu, The Introduction of Reaction-Diffusion Equations (in Chinese), Science Press, Beijing, 2011. Google Scholar

[30]

F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Equat., 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.  Google Scholar

[31]

J. Zhou and C. L. Mu, Coexistence of a diffusive predator-prey model with Holling type-Ⅱ functional response and density dependent mortality, J. Math. Anal. Appl., 385 (2012), 913-927.  doi: 10.1016/j.jmaa.2011.07.027.  Google Scholar

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